To answer both of your points, fast was implied by virtue of it being a communication network.

The open part can be pedantically removed in the case of ARPAnet, but I've not met anyone who confuses the Internet with ARPAnet. The Internet, as it came to be called in the 90's with the rise of the World Wide Web, WAS at it's core 'open'. Pretty much everyone I've met who was around and working on the ARPAnet saw it as a foregone conclusion it (a network based on the lessons learned through ARPAnet) would be going public in one way or another.